Animated Solution for Chemistry - Chemical Thermodynamics: In an irreverible process taking place at constant $T$ and $p$ and in which only pressure-volume work is being done, the change in Gibbs free energy ($dG$) and change in entropy ($dS$), satisfy the criteria

Understanding Irreversibility

Imagine dropping a drop of ink into a glass of clear water. Slowly, the ink diffuses, spreading out until the entire glass is a uniform shade of blue. Now, imagine waiting for the ink molecules to spontaneously gather back together into a single, perfect drop while the water becomes clear again. You could wait for the lifetime of the universe, and it would never happen.

This is the essence of an irreversible process. In thermodynamics, an irreversible process is one that occurs naturally in a specific direction and cannot reverse itself without external intervention. These processes are also called spontaneous processes. They are the physical manifestation of the "arrow of time," dictating the natural flow of events in our universe.

Entropy of the Universe

To quantify this natural directionality, physicists introduced the concept of entropy ($S$), which is often loosely described as a measure of disorder or randomness. The Second Law of Thermodynamics is arguably the most profound law in all of science. It states that for any spontaneous (irreversible) process, the total entropy of the universe must increase.

Mathematically, this is expressed as:

But calculating the entropy change of the entire universe is practically impossible. To make this law useful, we look at specific types of systems. Consider an isolated system—a system that cannot exchange either mass or energy with its surroundings. Because no heat is exchanged ($dq=0$), the surroundings are completely unaffected. Therefore, for an isolated system, the entropy of the system itself must increase for a process to be spontaneous.

An isolated system is characterized by having a constant Volume ($V$) and constant Internal Energy ($E$ or $U$). Thus, the thermodynamic criteria for spontaneity in an isolated system is written as:

This tells us that if we lock a system in a rigid, perfectly insulated box, any natural process that occurs inside will lead to an increase in its entropy.

Gibbs Free Energy

While the entropy criteria for an isolated system is elegant, it is not very practical for chemists. Most chemical reactions do not happen in rigid, insulated boxes. They happen in open beakers on laboratory benches. In these real-world scenarios, the system is exposed to the constant atmospheric pressure of the room and the constant temperature of the environment.

Under conditions of constant Temperature ($T$) and Pressure ($p$), the system exchanges heat with the surroundings, meaning the entropy of the surroundings changes. To avoid the tedious task of calculating the entropy change of the surroundings every time, Josiah Willard Gibbs introduced a new state function: Gibbs Free Energy ($G$).

Gibbs Free Energy is defined as:

Where $H$ is enthalpy, $T$ is absolute temperature, and $S$ is entropy. The brilliance of Gibbs Free Energy is that it packages the entropy change of the universe into a single property of the system alone.

By manipulating the Second Law of Thermodynamics for a system at constant $T$ and $p$, we find that the total entropy of the universe increases when the Gibbs Free Energy of the system decreases. Therefore, the criteria for a spontaneous (irreversible) process at constant temperature and pressure is that the change in Gibbs Free Energy must be negative:

This means the system naturally rolls down the "free energy hill" until it reaches a minimum, which corresponds to chemical equilibrium.

Returning to our original problem, we are asked to identify the correct criteria for an irreversible process. We have established two fundamental thermodynamic truths:

1. If the system were isolated (constant $V$ and $E$), its entropy must increase: $(dS{)}_{V,E}>0$. 2. If the system is kept at constant temperature and pressure, its Gibbs Free Energy must decrease: $(dG{)}_{T,p}<0$.

When we look at the given options, we see that option (b) perfectly aligns with both of these fundamental laws of nature. Understanding these criteria is not just about memorizing equations; it is about understanding the fundamental rules that govern every chemical reaction and physical transformation in the universe.